To determine the values of $p > 0$ for which the given series converges, we analyze the series:

$\sum_{n=1}^\infty \frac{\left( n^{1/3} + 1 \right)^p}{\left( n^{7/3} + n^{5/2} + 1 \right)^{13/11}}.$

Step-by-Step Analysis

1. Asymptotic Behavior:

   • For large $n$, the dominant term in $n^{1/3} + 1$ is $n^{1/3}$, and in $n^{7/3} + n^{5/2} + 1$, the dominant term is $n^{7/3}$.

   • Therefore, the series asymptotically behaves as: $\sum_{n=1}^\infty \frac{\left( n^{1/3} \right)^p}{\left( n^{7/3} \right)^{13/11}}.$

     Simplifying: $\sum_{n=1}^\infty \frac{n^{p/3}}{n^{(7/3) \cdot (13/11)}} = \sum_{n=1}^\infty \frac{1}{n^{\alpha}},$ where: $\alpha = \frac{7}{3} \cdot \frac{13}{11} - \frac{p}{3}.$

2. Convergence Criterion:

   • The series $\sum_{n=1}^\infty \frac{1}{n^\alpha}$ converges if and only if $\alpha > 1$.
   • Substituting $\alpha = \frac{7}{3} \cdot \frac{13}{11} - \frac{p}{3}$: $\frac{7}{3} \cdot \frac{13}{11} - \frac{p}{3} > 1.$

3. Simplify the Inequality:

   • Compute $\frac{7}{3} \cdot \frac{13}{11}$: $\frac{7 \cdot 13}{3 \cdot 11} = \frac{91}{33}.$

     • The inequality becomes: $\frac{91}{33} - \frac{p}{3} > 1.$

   • Subtract $1$ from both sides: $\frac{91}{33} - 1 > \frac{p}{3}.$

   • Simplify $\frac{91}{33} - 1 = \frac{91}{33} - \frac{33}{33} = \frac{58}{33}$: $\frac{58}{33} > \frac{p}{3}.$

   • Multiply through by $3$ to isolate $p$: $p < \frac{3 \cdot 58}{33} = \frac{174}{33}.$

4. Result:

   • The series converges if $0 < p < \frac{174}{33}$, or equivalently: $0 < p < \frac{58}{11}.$

Would you like further clarifications or have specific questions about the steps?

Related Questions:

1. What is the general criterion for the convergence of a $p$-series?
2. How does the dominant term affect the convergence of series?
3. Can we use comparison tests to verify the result?
4. How do fractional exponents impact series convergence?
5. What happens when $p > \frac{58}{11}$ or $p = \frac{58}{11}$?

Tip:

When determining the convergence of a series, always simplify the dominant terms for large $n$ to identify the growth rate accurately.